Trivially True and Implications

I have heard it said that an if-then statement is logically equivalent to a for all-statement.

For example the statement "If x is a natural number then x is a real number" can be rewritten as "For all x such that x is a natural number, x is a real number."

Then consider the statement "If x is a pink alligator then x has seven legs." It is clear that the hypothesis will always be false (because there are no pink alligators) and so the statement is considered trivially true.

Then its logical equivalent statement is: "For all pink alligators x, x has seven legs." And this statement would also be true.

Then consider the statement "If x is a pink alligator then x does not have seven legs." There are no pink alligators so the statement is trivially true, and so it is trivially true that for all pink alligators x, x does not have seven legs.

Then we have the statement, [for all x, P(x)] and [for all x, not P(x)]. I know this is not technically a case of A and NOT A both being true but I was wondering whether there is any other way to resolve this apparent paradox besides saying one statement is not the negation of the other.

I have heard it said that an if-then statement is logically equivalent to a for all-statement.

For example the statement "If x is a natural number then x is a real number" can be rewritten as "For all x such that x is a natural number, x is a real number."

Then consider the statement "If x is a pink alligator then x has seven legs." It is clear that the hypothesis will always be false (because there are no pink alligators) and so the statement is considered trivially true.

Then its logical equivalent statement is: "For all pink alligators x, x has seven legs." And this statement would also be true.

Then consider the statement "If x is a pink alligator then x does not have seven legs." There are no pink alligators so the statement is trivially true, and so it is trivially true that for all pink alligators x, x does not have seven legs.

Then we have the statement, [for all x, P(x)] and [for all x, not P(x)]. I know this is not technically a case of A and NOT A both being true but I was wondering whether there is any other way to resolve this apparent paradox besides saying one statement is not the negation of the other.

No paradox at all: from a false statement anything at all follows, in particular any other statement and its negation.

Interesting question. Yes, [for all x, P(x)] and [for all x, not P(x)] are both true. However, there is only one way to derive a contradiction, and that is to prove A and (not A) for some A. So, to derive a contradiction from these two statements, one must first eliminate the universal quantifier, i.e., instantiate x to some pink alligator. Since there are none, this is impossible.

For the argument above, it is not necessary to establish the connection between "if-then" and "for all". I would point out that "If x is a natural number then x is a real number" still has an implicit universal quantifier; otherwise, x is a free variable and this is not a proposition. But yes, is equivalent to .

In type theory, implication is indeed a special case of a universal quantifier. There, means "for every object x of type A, B", which is the same as "for every proof x of A, B". When x does not occur in B, this is just . However, this is probably beyond the scope of the question.